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For additive Gaussian white noise and a time-even order parameter, a midpoint discretization of the Onsager–Machlup functional evaluates drift derivatives at the midpoint configurations. Reversing a trajectory visits the same midpoints in reverse order, so the noise-to-path Jacobian determinant is invariant under reversal. It may depend on the trajectory, and must be distinguished from the constant normalization of the noise measure.

For nonconserved order-parameter dynamics with $\dot{p}=-\Gamma \mu [p]+f$, the factor at one time step, apart from a path-independent power of the step size, isHere the fields have first been restricted to a finite spatial grid, and ${\mathcal{H}}_n$ is the Hessian matrix of the free energy.

For mixed conserved and nonconserved order-parameter dynamics, the required noises for a joint configuration and flux trajectory arewhere $\mathsf{D},\mathsf{G}$ represent the divergence and gradient, and $\Delta =\mathsf{DG}$ represents the Laplacian. The Jacobian matrix isTaking its Schur complement gives the joint factorFor periodic boundary conditions, discretizing the two spatial operators compatibly gives $\mathsf{D}=-{\mathsf{G}}^{\mathsf{T}}$, so $\Gamma I-M\Delta$ is the positive relaxation operator when $\Gamma ,M>0$. Reversal leaves ${\mathcal{H}}_n$ unchanged and reverses the sign of the flux. The product of the joint factors is consequently the same for both histories, justifying its cancellation from the joint path probability of an order parameter and its flux. The functional chain rule used in the action ratio holds in the continuum limit of this midpoint convention.

For canonical positive kinetic terms in global supersymmetry, $V={\sum }_k|W_k{|}^2$ has mixed Hessian matrix $V_{i\bar{j}}={\sum }_k{W}_{ki}\overline{W_{kj}}$ and holomorphic block $V_{ij}={\sum }_k\overline{W_k}{W}_{kij}$. The $2n$ real scalar squared-mass sum is $2\mathrm{tr}(m^{†}m)$ for the chiral-superfield fermion mass matrix $m=W_{ij}$. Each Weyl spinor contributes two states, giving the same fermion weighted sum. Thus their boson-minus-fermion supertrace vanishes even with F-term breaking. The holomorphic Hessian splits real scalar masses without changing their sum. Noncanonical Kähler metrics, supergravity, and radiative corrections require different formulas.

A trust region restricts a local approximation to a neighborhood where it is expected to model the objective function accurately. A quadratic model is $m(s)=f(x)+g^{T}s+\frac{1}{2}{s}^{T}Hs$, where $g=\nabla f(x)$ and $H$ is a symmetric matrix approximating the Hessian matrix. Minimizing this quadratic function over a Euclidean ball of radius $\Delta >0$ is the trust-region subproblem. The trust-region acceptance ratio then checks the model against the actual objective function.

For a smooth real phase with an isolated nondegenerate interior critical point $q$, smooth amplitude $a$, and no competing boundary contribution, the leading contribution is $2\pi {\lambda }^{-1}a(q)|\det H{|}^{-1/2}{e}^{i\lambda \varphi (q)+i\pi \mathrm{sgn}(H)/4}$, where $H$ is the Hessian matrix. Orthogonal diagonalization gives two one-dimensional Fresnel integrals. A degenerate critical point or a stationary boundary needs its own local analysis.

A gradient integrand is uniformly convex in this elliptic variational sense when its Hessian matrix in the gradient variables has a common positive lower bound. Then $(F_p(x,p)-F_p(x,q))\cdot (p-q)\ge \lambda |p-q{|}^2$. Averaging $F_{pp}$ along a line segment in gradient space preserves the same lower bound.

For $D=\{y^{2}z=x^3\}$, the determinant of the homogeneous Hessian matrix is $24x{y}^2$. Its intersections with $D$ are the cusp $[0:0:1]$ and the smooth point $[0:1:0]$. The Hessian criterion for a flex therefore gives a unique smooth flex, with tangent $z=0$ and contact order three. Consequently every pair consisting of a triple line and a cuspidal cubic meeting only at a smooth point is projectively equivalent to $(3\{z=0\},D)$.